Large deviations for the largest eigenvalues and eigenvectors of spiked Gaussian random matrices

We consider matrices formed by a random N x N matrix drawn from the Gaussian Orthogonal Ensemble (or Gaussian Unitary Ensemble) plus a rank-one perturbation of strength theta, and focus on the largest eigenvalue, x, and the component, u, of the corresponding eigenvector in the direction associated to the rank-one perturbation. We obtain the large deviation principle governing the atypical joint fluctuations of x and u. Interestingly, for theta > 1, large deviations events characterized by a small value of u, i.e. u < 1 - 1/theta, are such that the second-largest eigenvalue pops out from the Wigner semi-circle and the associated eigenvector orients in the direction corresponding to the rank-one perturbation. We generalize these results to the Wishart Ensemble, and we extend them to the first n eigenvalues and the associated eigenvectors.